In this paper we assume that patients have preferences over potential donors, represented by ranks on in-going arcs, and aim at finding the maximum cardinality stable exchange. A stable exchange is a set of vertex-disjoint cycles such that there is no so-called blocking cycle where all the vertices would be better off, according to their preferences, than in the current solution. If strict preference is required only for one vertex in the blocking cycle then we speak about strongly stable exchanges. Stability and strong stability are equivalent to the core and strong core properties, respectively, which are standard solution concepts in cooperative game theory, see [1]} and [2]}. The usage of stable matchings in two-sided matching markets are standard both in theory and practice, in applications such as resident allocation, college admission and school choice, see e.g. a recent book on the algorithmic aspects of this topic by [3]}.
